#!/usr/bin/env python3
"""
Computation 87 -- Bernoulli Laplace-transform identification for Z^2: Goldstone theta-phase Laplace-transform
========================================================================
Bernoulli Laplace-transform identification: identify Z^2 not via CC inner fluctuation but via
the asymptotic Laplace transform of the substrate Bernoulli measure,
exploiting the closed Goldstone identification theta = H^0 phase.

GOLDSTONE BACKGROUND
--------------------
The Goldstone-mode item (closed v23.41-42) identifies the substrate's
theta phase with the Higgs neutral-component phase H^0.  Concretely:
the Higgs field H = (H^+, H^0) has neutral component H^0 = (v + h)/sqrt(2)
* exp(i theta), with theta the Goldstone mode eliminated by gauge fixing
(Stueckelberg).

If Z^2 is somehow the LAPLACE TRANSFORM of the substrate empirical mean
evaluated at a specific scale, the e^{-1} value emerges naturally.

CALCULATION
-----------
Substrate Bernoulli measure mu = otimes Bern(1/2) on configurations
C in P(D).  Empirical mean of substrate bits:
  X_bar = (1/D) sum_a 1[a in C]

Under mu, X_bar has mean 1/2 and variance 1/(4D) (Bernoulli LLN).

Laplace transform of X_bar:
  E[exp(c X_bar)] = ((1 + exp(c/D)) / 2)^D
                 -> exp(c / 2)   as D -> infty   (by CLT / Cramer)

For Z^2 = e^{-1}, we need exp(c / 2) = e^{-1}, i.e. c = -2.

THE QUESTION
------------
Is c = -2 structurally natural from the substrate measure?

Possible readings:
  (a) c = -2 = -1 / mu_site  (since mu_site = 1/2)
  (b) c = -2 = -4 sigma^2  (since sigma^2 = mu_site(1-mu_site) = 1/4
                            so 4 sigma^2 = 1, and the -2 sign carries
                            an extra factor of 2 from variance doubling)
  (c) c = -2 = number of substrate states per bit  (Z_2 = {0, 1})
  (d) c = -2 = "two-sigma" tilting parameter for asymptotic large
              deviations

None of (a)-(d) is uniquely forced by P1 (which only delivers
mu_site = 1/2, not the specific coefficient -2 of the Laplace
transform argument).

Conclusion
-------
The Laplace-transform identification Z^2 = E[exp(c X_bar)] with c = -2
DOES deliver e^{-1} asymptotically -- but only IF we accept c = -2 as
the natural scale.  Without an independent structural derivation of
c = -2 from P1 alone, this identification is form-compatible but not
directly derived.

The candidate readings (a)-(d) all have structural flavour, but none
is uniquely forced.  This is analogous to the Yukawa coupling y/Lambda
= 2^{-1/4} in Angle A: the right scaling form exists but the specific
multiplicative constant escapes derivation from P1 alone.

CONCLUSION
----------
Angle D delivers e^{-1} as a Laplace-transform value but does not close
Z^2 = e^{-1} as a structural identity, because the required scale c = -2
is not uniquely forced by P1.

Together with Comps 84, 85, 86 (Angles A, B, C), Comp 87 confirms:
The four CC-inner-fluctuation and Laplace-transform approaches of Comps 84-87 each leave the multiplicative constant un-derived without an additional structural input; Comp 88 supplies that input via the KO-dimension reading.
The substrate-side e^{-1} is structurally exact (Comp 62), the SM-side
Z^2 is 0.8% near-coincidence, and the bridge between them remains
genuinely open.
"""

from __future__ import annotations
import math
import numpy as np


def main():
    print("=" * 100)
    print("  Computation 87 -- Bernoulli Laplace-transform identification for Z^2: Laplace-transform identification")
    print("=" * 100)
    print()

    print("LAPLACE TRANSFORM OF BERNOULLI EMPIRICAL MEAN")
    print("-" * 100)
    print("  E[exp(c X_bar)] = ((1 + exp(c/D)) / 2)^D")
    print()
    print(f"  {'D':>6}  {'c = -1':>15}  {'c = -2':>15}  {'c = -2 limit':>18}")
    for D in [10, 100, 1000, 10000]:
        for c in [-1, -2]:
            val = ((1 + math.exp(c / D)) / 2) ** D
        c_minus2 = -2
        val_minus2 = ((1 + math.exp(c_minus2 / D)) / 2) ** D
        val_minus1 = ((1 + math.exp(-1 / D)) / 2) ** D
        print(f"  {D:>6}  {val_minus1:>15.6f}  {val_minus2:>15.6f}  "
              f"{abs(val_minus2 - math.exp(-1)):>15.2e}")
    print()
    print(f"  Asymptotic limit (D -> infty):  E[exp(c X_bar)] -> exp(c / 2)")
    print(f"  For c = -2:  limit = exp(-1) = {math.exp(-1):.6f}")
    print(f"  For c = -1:  limit = exp(-1/2) = {math.exp(-0.5):.6f}")
    print()

    print("THE STRUCTURAL READING FOR c = -2")
    print("-" * 100)
    print()
    print("  Possible structural readings of c = -2:")
    print("    (a) c = -1 / mu_site  =  -1 / (1/2)  =  -2")
    print("    (b) c = -4 sigma^2 * (factor 2)  =  -4 * (1/4) * 2  =  -2")
    print("    (c) c = -(number of substrate states per bit)  =  -2  (Z_2)")
    print("    (d) c = -2-sigma tilting parameter for asymptotic LD")
    print()
    print("  Each reading is structurally suggestive but NONE is uniquely")
    print("  forced by P1 (which delivers mu_site = 1/2 as the only")
    print("  parameter).  P1 does not specify which functional of mu_site")
    print("  the Laplace transform argument should equal.")
    print()

    print("Conclusion (HONEST)")
    print("-" * 100)
    print()
    print("  The Laplace transform formalism delivers e^{-1} at c = -2.")
    print("  But the choice c = -2 is form-compatible with multiple structural")
    print("  readings, none uniquely forced.  Without an independent argument")
    print("  for c = -2 from P1 alone, Z^2 = E[exp(-2 X_bar)] is a")
    print("  recognition rather than a derivation.")
    print()
    print("  This is analogous to the Yukawa-coupling situation in Angle A:")
    print("  the right SCALING FORM (y/Lambda = sqrt(p) for Bernoulli p) is")
    print("  available, but the specific p = 1/sqrt(2) is not uniquely forced.")
    print()
    print("  The Laplace-transform identification at this level does not deliver structural closure; Comp 88 sharpens it via the KO-dim reading.")
    print()


if __name__ == "__main__":
    main()